An Asymptotic Approach to Twin Primes in a Prime-Dependent Interval

The Twin Prime Conjecture is a hypothesis that asserts the existence of infinitely many prime pairs of the form (p, p + 2) and is considered one of the most longstanding open problems in analytic number theory. In this study, we present a new approach to proving the infinitude of twin primes by examining specific prime number intervals. In particular, we demonstrate that there must be at least one twin prime pair within the interval (pkpk+1, p2k+1). First, we prove that there is at least one prime in this interval using the Prime Counting Function, and then we establish that there are at least two primes. Subsequently, by employing the Hardy-Littlewood estimate and the Montgomery-Vaughan relations, we mathematically prove that at least one of these primes must form a twin prime pair. Additionally, we analyze the error term in this study, showing that the ratio of the error term to the main integral approaches zero, thereby ensuring the robustness of the obtained result.